FOM: second-order logic is a myth

Pat Hayes phayes at
Wed Mar 10 12:15:29 EST 1999

Charlie Silver writes:

>	To me, the major point that separates (intuitive) quantification
>from set theory is that you can say "for all x..." without implying that
>the x's must be *in* something.  For example, take: "All Canada geese fly
>south for the winter."  I don't think anything in this statement implies
>that in addition to there being some number of geese there is also a *set*
>of geese.

Surely it isn't proper to think of a set as a container. For example, if I
remove something from a container it stays the same, but removing something
from a set gets you a different set. (The set is everything that is 'in'
the 'container', not the container itself.)

In contrast to Silver's intuition, it seems to me that saying "for all
x..." implies that one has some notion of the extent over which that "all"
is supposed to reach. If I can't (even conceptually) distinguish a Canada
goose from a hooting swan, then for me to say "All Canada geese..." is
impossibly vague. (How can we evaluate a proposed counterxample if we can't
know whether or not is a goose?) The meaning is indeterminate until we
decide what counts as a Canada goose. But that amounts to deciding what is
in the set of Canada geese. The point is that the criteria which determine
the truthvalue of "for all x..", and those which make all the x's into a
set, are both purely extensional; and they coincide; so they are the same.

Pat Hayes

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